locus in equilateral , about distances from it's sides

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jlacosta

Apr 2, 2025

0 replies

For positive integers $ a, b, c $, find all possible positive integer values o

Jackson0423 9

N 11 minutes ago by SpeedCuber7

For positive integers $a, b, c$ , find all possible positive integer values of
$\frac{a}{b} + \frac{b}{c} + \frac{c}{a}.$

9 replies

Jackson0423

Apr 13, 2025

SpeedCuber7

11 minutes ago

24 Aug FE problem

nicky-glass 3

N 14 minutes ago by pco

Source: Baltic Way 1995

$f:\mathbb R\setminus \{0\} \to \mathbb R$
(i) $f(1)=1$ ,
(ii) $\forall x,y,x+y \neq 0:f(\frac{1}{x+y})=f(\frac{1}{x})+f(\frac{1}{y}) : P(x,y)$
(iii) $\forall x,y,x+y \neq 0:(x+y)f(x+y)=xyf(x)f(y) :Q(x,y)$
$f=?$

3 replies

nicky-glass

Aug 24, 2016

pco

14 minutes ago

Simply equation but hard

giangtruong13 1

N 14 minutes ago by anduran

Find all integer pairs $(x,y)$ satisfy that: $(x^2+y)(y^2+x)=(x-y)^3$

1 reply

1 viewing

giangtruong13

2 hours ago

anduran

14 minutes ago

Number Theory

AnhQuang_67 0

19 minutes ago

Find all pairs of positive integers $(m,n)$ satisfying $2^m+21^n$ is a perfect square

0 replies

AnhQuang_67

19 minutes ago

0 replies

Congrats to the San Diego Math Circle

rrusczyk 0

Apr 15, 2010

One difficulty facing many Math Circles is the sad fact that most don't survive their founder leaving. At gatherings of Math Circle directors, I have been known to gloat that the San Diego Math Circle is one of the few that hasn't had this problem, because for a couple years now, the San Diego Math Circle has been run entirely by the parents, mainly David Brown. Not only has the Math Circle continued, but it has thrived. Congrats to all the students, and to all the parents who continue to make the Math Circle so special.

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rrusczyk

Apr 15, 2010

0 replies

No more topics!

locus in equilateral , about distances from it's sides

parmenides51 2

N Sep 23, 2017 by richrow12

Source: Nordic Mathematical Contest 2014 #2

Given an equilateral triangle, find all points inside the triangle such that the distance from the point to one of the sides is equal to the geometric mean of the distances from the point to the other two sides of the triangle.

2 replies

parmenides51

Sep 23, 2017

richrow12

Sep 23, 2017

locus in equilateral , about distances from it's sides

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Source: Nordic Mathematical Contest 2014 #2

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parmenides51

30629 posts

parmenides51

#1 h Sep 23, 2017, 12:41 PM • 1 Y

Y by Adventure10

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richrow12

411 posts

richrow12

#2 h Sep 23, 2017, 4:21 PM • 2 Y

Y by Adventure10, Mango247

I get answer.
Answer

Let $ABC$ be an original equilateral triangle with center at point $O$ .
Then, answer is all points $P$ , satysfying at least one of three conditions $\angle APB=120^{\circ}$ or $\angle BPC=120^{\circ}$ or $\angle CPA=120^{\circ}$ .

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richrow12

411 posts

richrow12

#3 h Sep 23, 2017, 4:31 PM • 1 Y

Y by Adventure10

Solution

Denote distance from $P$ to line $\ell$ as $\rho(P,\ell)$ .
Suppose that $\rho^2(P,AB)=\rho(P,AC)\rho(P,BC)$ . We will prove that $\angle APB = 120^{\circ}$ .
Since $AB=AC=BC$ we conclude that $S^2_{APB}=S_{APC}S_{BPC}$ . Let $A_1=AP~\cap~BC$ and $B_1=BP~\cap~AC$ .
Observe that $\frac{BA_1}{CA_1}=\frac{S_{APB}}{S_{APC}}$ and $\frac{AB_1}{CB_1}=\frac{S_{APB}}{S_{BPC}}$ .
Hence, $\frac{BA_1}{CA_1}=\frac{CB_1}{AB_1}$ . From this we get $BA_1=CB_1$ and $A_1C=B_1A$ . Therefore $\angle APB=120^{\circ}$ .

For such points equality $\rho^2(P,AB)=\rho(P,AC)\rho(P,BC)$ can easy proved.

This post has been edited 1 time. Last edited by richrow12, Sep 23, 2017, 4:32 PM

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